## Real World Applications – Algebra I

### Topic

How can we lower workers’ compensation in companies using Algebra?

### Student Exploration

Most companies have a Human Resources department that handles all of the logistics about the company’s workers. One of their biggest responsibilities is working with employees when they get hurt on the job. When this happens, the company offers workers’ compensation. For one particular company, the human resources advisor created algebraic relationships to represent the number of people that were hurt at one of the branches of the company. He concluded that the function that represents this relationship is \begin{align*}I_0(x)= \frac{500}{x}\end{align*}

We can represent this revised model by taking the difference between the two relationships as one rational expression.

\begin{align*}I_0-I_1= \frac{500}{x} - \frac{500}{x+1}\end{align*}

Since we’re subtracting two rational expressions, we have to make sure we have a common denominator, and then subtract. In this case, our common denominator is \begin{align*}x(x + 1)\end{align*}

In the picture above, we found the common denominator by making the denominators the same, and then we simplified the numerator.

What does this resulting expression mean?

We can input some values into a table for this function and make graph the relationship. It would look like this:

For this graph, the \begin{align*}x\end{align*}

The graphs are really close together, so it’s a little difficult to compare. So, let’s take a look at the tables that represent the two relationships.

# of hours of training \begin{align*}(x)\end{align*} |
Without plan | With plan |
---|---|---|

5 | 100 | 83.33333333 |

10 | 50 | 45.45454545 |

15 | 33.33333333 | 31.25 |

20 | 25 | 23.80952381 |

25 | 20 | 19.23076923 |

30 | 16.66666667 | 16.12903226 |

50 | 10 | 9.803921569 |

100 | 5 | 4.95049505 |

200 | 2.5 | 2.487562189 |

Looking at this table, we can see that the plan will definitely make a difference. For every number of hours of training, we can see that the rightmost column has significantly lower numbers. The plan will work!

We can also use the formula \begin{align*}I_0-I_1= \frac{500}{x}- \frac{500}{x+1}\end{align*}

Let’s use \begin{align*}\frac{500}{x^2+x}\end{align*}

When we set this equal to 5, we can look at solving this two different ways. We can look at this in the sense like we’re solving a proportion and 5 is the same as \begin{align*}\frac{5}{1}\end{align*}

\begin{align*}\frac{500}{x^2+x}=\frac{5}{1}\end{align*}

Or, we can solve it like we’re rationalizing the equation and trying to clear the denominator in this rational equation. We would take \begin{align*}\frac{500}{x^2+x}=5\end{align*}

Let’s work with the proportion \begin{align*}\frac{500}{x^2+x}=\frac{5}{1}\end{align*}

After using the quadratic formula, we’ve found two answers. Would both answers make sense in this situation? And what do these answers mean?

Since we’re trying to find out how many trainings it would take to ensure that the number of incidents are no more than 5, 9.5 would make sense. Offering -10.5 trainings wouldn’t make sense!

### Extension Investigation

Research different companies that would represent a similar type of model. Why would this company have this type of model? Why did you choose this company? Would this model be limited to injuries? What about production of certain goods?